The dice that felt wrong

Ask thirty students to write down a made-up sequence of forty coin tosses, and ask thirty more to toss a coin forty times and record the result. A teacher can usually tell the two groups apart in under a minute.

The invented sequences alternate too much. People write heads and tails in short alternating bursts and avoid long runs, because a run of five heads feels like a pattern and patterns feel unlike randomness. Real forty-toss sequences contain a run of four or more about three quarters of the time, and a run of five is unremarkable. The invented ones almost never have them.

What this reveals is not innumeracy but a specific wrong model. People treat chance as self-correcting, as though a coin that has come up heads three times owes a tail. The coin has no memory, and the correct statement is the other way round: in a long sequence, every short pattern you can name will eventually appear, including the ones that look designed.

The same intuition misleads in more serious places. A run of four dry years is read as making rain due. A pattern in a small set of medical cases is read as a cluster with a cause. Both can be real and both are what pure chance looks like at that scale, which is why the first question about any apparent cluster is how many opportunities there were for one to appear somewhere.

The exercise is worth doing on yourself once. Write forty tosses, then count your longest run. If it is three, you have just demonstrated the bias in the most direct way available, and the lesson sticks in a way that reading about it does not.